Let σm=n∑r=1rm.
Refer to the tabulation of the power sum of integers here.
It is interesting to note that
σ1 =12n(n+1)σ2 =16n(n+1)(2n+1)σ3 =14n2(n+1)2=σ12σ4 =130n(n+1)(2n+1)(3n2+3n−1)=15σ2 (3n2+3n−1)σ5 =112n2(n+1)2(2n2+2n−1)=13σ3 (2n2+2n−1)σ6 =142n(n+1)(2n+1)(3n4+6n3−3n+1)=17σ2 (3n4+6n3−3n+1)σ7 =124n2(n+1)2(⋯)=16σ3 (⋯)σ8 =190n(n+1)(2n+1)(⋯)=115σ2 (⋯)σ9 =120n2(n+1)2(n2+n−1)(⋯)=15σ3 (n2+n−1)(⋯)σ10=166n(n+1)(2n+1)(n2+n−1)(⋯)=111σ2 (n2+n−1)(⋯)
i.e.
- the sum of squares, σ2, is a factor of sum of even powers greater than 2, and
- the sum of cubes, σ3, is a factor of sum of odd powers greater than 3.
Is there a simple explanation for this, if possible without using Faulhaber's formula and Bernoulli numbers, etc?
and also,
Why does this occur only for σ2,σ3 but not for σ4,σ5, etc?
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